2: Allele and Genotype Frequencies
In this chapter we will work through how the basics of Mendelian genetics play out at the population level in sexually reproducing organisms.
Loci and alleles are the basic currency of population genetics–and indeed of genetics. A locus may be an entire gene, or a single nucleotide base pair such as A-T. At each locus, there may be multiple genetic variants segregating in the population—these different genetic variants are known as alleles . If all individuals in the population carry the same allele, we say that the locus is monomorphic ; at this locus there is no genetic variability in the population. If there are multiple alleles in the population at a locus, we say that this locus is polymorphic (this is sometimes referred to as a segregating site).
|
pos. |
con. |
a |
b |
c |
d |
e |
f |
g |
h |
i |
j |
k |
l |
a |
b |
c |
d |
e |
f |
a |
b |
c |
d |
e |
f |
g |
h |
i |
j |
k |
l |
NS/S |
||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
781 |
G |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
NS |
||
|
789 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
S |
||
|
808 |
A |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
NS |
||
|
816 |
G |
T |
T |
T |
T |
- |
- |
- |
- |
- |
- |
- |
T |
T |
T |
T |
T |
T |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
834 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
- |
- |
- |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
859 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
G |
NS |
||
|
867 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
G |
G |
G |
G |
G |
A |
G |
G |
G |
G |
G |
G |
S |
||
|
870 |
C |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
950 |
G |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
974 |
G |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
T |
- |
T |
T |
T |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
983 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
S |
||
|
1019 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
1031 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
- |
- |
- |
S |
||
|
1034 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
C |
C |
C |
- |
- |
C |
- |
C |
C |
S |
|||
|
1043 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
1068 |
C |
T |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
1089 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
A |
A |
A |
A |
A |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
NS |
||
|
1101 |
G |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
A |
NS |
||
|
1127 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
S |
||
|
1131 |
C |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
T |
- |
- |
- |
- |
- |
- |
- |
S |
||
|
1160 |
T |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
C |
S |
Table \(\PageIndex{1}\) shows a small stretch of orthologous sequence for the ADH locus from samples from Drosophila melanogaster , D. simulans , and D. yakuba . D. melanogaster and D. simulans are sister species and D. yakuba is a close outgroup to the two. Each column represents a single haplotype from an individual (the individuals are diploid but were inbred so they’re homozygous for their haplotype). Only sites that differ among individuals of the three species are shown. Site \(834\) is an example of a polymorphism; some D. simulans individuals carry a \(C\) allele while others have a \(T\) . Fixed differences are sites that differ between the species but are monomorphic within the species. Site \(781\) is an example of a fixed difference between D. melanogaster and the other two species.
We can also annotate the alleles and loci in various ways. For example, position \(781\) is a non-synonymous fixed difference. We call the less common allele at a polymorphism the minor allele and the common allele the major allele , e.g. at site \(1068\) the \(T\) allele is the minor allele in D. melanogaster . We call the more evolutionarily recent of the two alleles the derived allele and the older of the two the ancestral allele . We infer that the \(T\) allele at site 1068 is the derived allele because the \(C\) is found in both other species, suggesting that the \(T\) allele arose via a \(C \rightarrow T\) mutation.
- How many segregating sites does the sample from D. simulans have in the ADH gene?
- How many fixed differences are there between D. melanogaster and D. yakuba ?
Allele frequencies
Allele frequencies are a central unit of population genetics analysis, but from diploid individuals we only get to observe genotype counts. Our first task then is to calculate allele frequencies from genotype counts. Consider a diploid autosomal locus segregating for two alleles ( \(A_1\) and \(A_2\) ). We’ll use these arbitrary labels for our alleles, merely to keep this general. Let \(N_{11}\) and \(N_{12}\) be the number of \(A_1A_1\) homozygotes and \(A_1A_2\) heterozygotes, respectively. Moreover, let \(N\) be the total number of diploid individuals in the population. We can then define the relative frequencies of \(A_1A_1\) and \(A_1A_2\) genotypes as \(f_{11} = N_{11}/N\) and \(f_{12} = N_{12}/N\) , respectively. The frequency of allele \(A_1\) in the population is then given by
\[p = \frac{2 N_{11} + N_{12}}{2N} = f_{11} + \frac{1}{2} f_{12}. \nonumber\]
Note that this follows directly from how we count alleles given individuals’ genotypes, and holds independently of Hardy–Weinberg proportions and equilibrium (discussed below). The frequency of the alternate allele ( \(A_2\) ) is then just \(q=1-p\) .
Measures of genetic variability
Nucleotide diversity (\(\pi\)) - One common measure of genetic diversity is the average number of single nucleotide differences between haplotypes chosen at random from a sample. This is called nucleotide diversity and is often denoted by \(\pi\) . For example, we can calculate \(\pi\) for our ADH locus from Table \ref{Table:ADH} above: we have 6 sequences from D. simulans (a-f), there’s a total of 15 ways of pairing these sequences, and
\[\pi=\frac{1}{15} \big( (2 + 1 + 1 + 1 + 0 ) + (3 + 3 + 3 + 2 ) +(0 + 0 + 1) + (0 + 1) + (1) \big)=1.2\overline{6}\]
where the first bracketed term gives the pairwise differences between a and b-f, the second bracketed term the differences between b and c-f and so on.
Our \(\pi\) measure will depend on the length of sequence it is calculated for. Therefore, \(\pi\) is usually normalized by the length of sequence, to be a per site (or per base) measure. For example, our ADH sequence covers \(397\) bp of DNA and so \(\pi = 1.2\overline{6}/397=0.0032\) per site in D. simulans for this region. Note that we could also calculate \(\pi\) per synonymous site (or non-synonymous). For synonymous site \(\pi\) , we would count up number of synonymous differences between our pairs of sequences, and then divide by the total number of sites where a synonymous change could have occurred. Technically we would need to divide by the total number of possible point mutations that would result in a synonymous change; this is because some mutational changes at a particular nucleotide will result in a non-synonymous or synonymous change depending on the base-pair change.
Number of segregating sites
Another measure of genetic variability is the total number of sites that are polymorphic (segregating) in our sample. One issue is that the number of segregating sites will grow as we sequence more individuals (unlike \(\pi\) ). Later in the course, we’ll talk about how to standardize the number of segregating sites for the number of individuals sequenced (see \ref{watterson_theta}).
The frequency spectrum
We also often want to compile information about the frequency of alleles across sites. We call alleles that are found once in a sample singletons , alleles that are found twice in a sample doubletons , and so on. We count up the number of loci where an allele is found \(i\) times out of \(n\) , e.g. how many singletons are there in the sample, and this is called the frequency spectrum . We’ll want to do this in some consistent manner, such as calculating the frequency spectrum of the minor allele or the derived allele.
How many minor-allele singletons are there in D. simulans in the ADH region? [Defining minor allele just within D. simulans .]
Levels of genetic variability across species
Two observations have puzzled population geneticists since the inception of molecular population genetics. The first is the relatively high level of genetic variation observed in most obligately sexual species. This first observation, in part, drove the development of the Neutral theory of molecular evolution, the idea that much of this molecular polymorphism may simply reflect a balance between genetic drift and mutation. The second observation is the relatively narrow range of polymorphism across species with vastly different census sizes. This observation represented a puzzle as the Neutral theory predicts that levels of genetic diversity should scale with population size. Much effort in theoretical and empirical population genetics has been devoted to trying to reconcile models with these various observations. We’ll return to discuss these ideas throughout our course.
The first observations of molecular genetic diversity within natural populations were made from surveys of allozyme data, but we can revisit these general patterns with modern data. For example, Leffler et al. (2012) compiled data on levels of within-population, autosomal nucleotide diversity (π) for 167 species across 14 phyla from non-coding and synonymous sites (Figure 2.3). The species with the lowest levels of π in their survey was Lynx, with π = 0.01%, i.e. only 1/10000 bases differed between two sequences. In contrast, some of the highest levels of diversity were found in Ciona savignyi , Sea Squirts, where a remarkable 1/12 bases differ between pairs of sequences. This 800-fold range of diversity seems impressive, but census population sizes have a much larger range.
Hardy–Weinberg proportions
Imagine a population mating at random with respect to genotypes, i.e. no inbreeding, no assortative mating, no population structure, and no sex differences in allele frequencies. The frequency of allele \(A_1\) in the population at the time of reproduction is \(p\) . An \(A_1A_1\) genotype is made by reaching out into our population and independently drawing two \(A_1\) allele gametes to form a zygote. Therefore, the probability that an individual is an \(A_1A_1\) homozygote is \(p^2\) . This probability is also the expected frequencies of the \(A_1A_1\) homozygote in the population. The expected frequency of the three possible genotypes are
| \(f_{11}\) | \(f_{12}\) | \(f_{22}\) |
|---|---|---|
| \(p^2\) | \(2pq\) | \(q^2\) |
i.e. their Hardy-Weinberg expectations . Note that we only need to assume random mating with respect to our focal allele in order for these expected frequencies to hold in the zygotes forming the next generation. Evolutionary forces, such as selection, change allele frequencies within generations, but do not change this expectation for new zygotes, as long as \(p\) is the frequency of the \(A_1\) allele in the population at the time when gametes fuse. We only need the assumptions of no migration, selection, and mutation in order for these Hardy-Weinberg expectations of genotypes to represent a long term equilibrium.
On the coastal islands of British Columbia there is a subspecies of black bear ( Ursus americanus kermodei , Kermode’s bear). Many members of this black bear subspecies are white; they’re sometimes called spirit bears. These bears aren’t hybrids with polar bears, nor are they albinos. They are homozygotes for a recessive change at the MC1R gene. Individuals who are \(GG\) at this SNP are white, while \(AA\) and \(AG\) individuals are black.
Below are the genotype counts for the MC1R polymorphism in a sample of bears from British Columbia’s island populations from .
| \(AA\) | \(AG\) | \(GG\) |
|---|---|---|
| 42 | 24 | 21 |
What are the expected frequencies of the three genotypes under HW?
See Figure \ref{fig:HWE_CEU_YRI} for a nice empirical demonstration of Hardy–Weinberg proportions. The mean frequency of each genotype closely matches its HW expectations, and much of the scatter of the dots around the expected line is due to our small sample size ( \(\sim 60\) individuals). While HW often seems like a silly model, it often holds remarkably well within populations. This is because individuals don’t mate at random, but they do mate at random with respect to their genotype at most of the loci in the genome.
You are investigating a locus with three alleles, A, B, and C, with allele frequencies \(p_A\) , \(p_B\) , and \(p_C\) . What fraction of the population is expected to be homozygotes under Hardy–Weinberg?
Microsatellites are regions of the genome where individuals vary for the number of copies of some short DNA repeat that they carry. These regions are often highly variable across individuals, making them a suitable way to identify individuals from a DNA sample. This so-called DNA fingerprinting has a range of applications from establishing paternity and identifying human remains to matching individuals to DNA samples from a crime scene. The FBI make use of the CODIS database. The CODIS database contains the genotypes of over 13 million people, most of whom have been convicted of a crime. Most of the profiles record genotypes at 13 microsatellite loci that are tetranucleotide repeats (since 2017, 20 sites have been genotyped).
The allele counts for two loci (D16S539 and TH01) are shown in Table \(\PageIndex{2}\) and Table \(\PageIndex{3}\) for a sample of 155 people of European ancestry. You can assume these two loci are on different chromosomes.
| allele name | 80 | 90 | 100 | 110 | 120 | 121 | 130 | 140 | 150 |
| allele count | 3 | 34 | 13 | 102 | 97 | 1 | 44 | 13 | 3 |
Table \(\PageIndex{3}\): Same as Table \(\PageIndex{2}\) but for the TH01 microsatellite.
| allele name | 60 | 70 | 80 | 90 | 93 | 100 | 110 |
| allele count | 84 | 42 | 37 | 67 | 77 | 1 | 2 |
You extract a DNA sample from a crime scene. The genotype is 100/80 at the D16S539 locus and 70/93 at TH01.
- You have a suspect in custody. Assuming this suspect is innocent and of European ancestry, what is the probability that their genotype would match this profile by chance (a false-match probability)?
- The FBI uses \(\geq\) 13 markers. Why is this higher number necessary to make the match statement convincing evidence in court?
- An early case that triggered debate among forensic geneticists was a crime among the Abenaki, a Native American community in Vermont . There was a DNA sample from the crime scene, and the perpetrator was thought likely to be a member of the Abenaki community. Given that allele frequencies vary among populations, why would people be concerned about using data from a non-Abenaki population to compute a false match probability?
Assortative mating
One major violation of the assumptions of Hardy Weinberg is non-random mating with respect to the genotype at a locus. One way that individuals can mate non-randomly is if individuals choose to mate based on a phenotype determined by (in part) the genotype at a locus. This non-random mating can be between: 1) individuals with similar phenotype, so called positive assortative mating or 2) individuals with dissimilar phenotypes, negative assortative mating or disassortative mating. Here we’ll briefly discuss a couple of real examples of assortative mating to make sure we’re all on the same page. We’ll encounter other forms of non-random mating, due to inbreeding and population structure, in the next few chapters.
Positive assortative mating on the basis of a phenotype can create an excess of homozygotes. Heliconius butterflies are famous for their mimicry, where poisonous pairs of distantly related species mimic each others’ bright colour patterns and so share the benefits of being avoided by visual predators (Müllerian mimics). H. melpomene rosina and H. cydno chioneus are closely related species that co-occur in central Panama, but mimic different other co-occuring species (Figure \(\PageIndex{6}\) ). These differences in colouration pattern are due to a few loci with large phenotypic effects. The two species can hybridize and produce viable F1 hybrids. These F1 hybrids are heterozygotes at the colour loci, and their intermediate appearance means that they’re poor mimics and so are quickly eaten by predators. However, these heterzygote (F1) hybrids are very rare in nature Figure \(\PageIndex{7}\), as the parental species show strong positive assortatively mating based on colour pattern, based on genetic differences in mate preference .
Disassortative mating, mating of unlike individuals, can lead to an excess of heterozygotes and a deficit of homozygotes. One example of very strong disassortative mating is offered by white-throated sparrows ( Zonotrichia albicollis ). In white-throated sparrows, there is a white-striped and a tan-striped morph, with female and male white-striped morphs have a much brighter white stripe and throat. There is very strong disassortative mating in this system, with 1099 out of 1116 nesting pairs consisting of one tan- and one white-striped morph and only 17 of these nesting pairs being different morphs . The difference between these morphs has a simple inheritance pattern, with white being due to a single dominant allele (called 2m) and tan color from a recessive allele called 2. Thus strong disassortative mating has a strong effect on the genotype frequencies:
| Tan | White | (Super)White |
|---|---|---|
| 2/2 | 2/2m | 2m/2m |
| 978 | 1011 | 3 |
There are almost no 2m homozygotes (so called Super white individuals) despite the 2m allele being common in the population (data from Tuttle et al., 2016, table S1).
Another important example of disassortative mating are mating type systems, which are present in many fungi, algae, and protozoa. Gametes of the same species can only fuse to form a zygote if they differ in mating type. The mating type of gametes is genetically controled by a mating type locus, and so individuals are nearly always heterozygous at this locus. In some groups of organisms, there are just two different alleles, in other clades these loci have tens or hundreds of alleles.
Allele sharing among related individuals and Identity by Descent
All of the individuals in a population are related to each other by a giant pedigree (family tree). For most pairs of individuals in a population these relationships are very distant (e.g. distant cousins), while some individuals will be more closely related (e.g. sibling/first cousins). All individuals are related to one another by varying levels of relatedness, or kinship . Related individuals can share alleles that have both descended from the shared common ancestor. To be shared, these alleles must be inherited through all meioses connecting the two individuals (e.g. surviving the \(\frac{1}{2}\) probability of segregation each meiosis). As closer relatives are separated by fewer meioses, closer relatives share more alleles. In Figure \(\PageIndex{10}\) we show the sharing of chromosomal regions between two cousins. As we’ll see, many population and quantitative genetic concepts rely on how closely related individuals are, and thus we need some way to quantify the degree of kinship among individuals.
We will define two alleles to be identical by descent (IBD) if they are identical due to transmission from a common ancestor in the past few generations. For the moment, we ignore mutation, and we will be more precise about what we mean by ‘past few generations’ later on. For example, parent and child share exactly one allele identical by descent at a locus, assuming that the two parents of the child are randomly mated individuals from the population. In Figure \(\PageIndex{16}\), I show a pedigree demonstrating some configurations of IBD.
One summary of how related two individuals (let’s call them \(i\) and \(j\) ) are is the probability that our pair of individuals share 0, 1, or 2 alleles identical by descent (see Figure \(\PageIndex{11}\)). We denote these identity-by-descent probabilities by \(r_0\) , \(r_1\) , and \(r_2\) respectively. See Table \(\PageIndex{4}\) for some examples. We can also interpret these probabilities as genome-wide averages. For example, on average, at a quarter of all their autosomal loci full-sibs share zero alleles identical by descent.
One summary of relatedness that will be important is the probability that two alleles (I & J) picked at random, one from each of the two different individuals \(i\) and \(j\) , are identical by descent ( \(P(\text{i & j IBD})\) ). We call this quantity the coefficient of kinship of individuals \(i\) and \(j\) , and denote it by \(F_{ij}\) . It is calculated as
\[\begin{aligned} F_{ij} = & \mathbb{P}(\text{I & J IBD} )\\[4pt] =& \mathbb{P}(\text{I & J IBD} |~ \text{i & j 0 IBD}) \mathbb{P}(\text{i & j 0 IBD}) + \mathbb{P}(\text{I & J IBD} |~ \text{i & j 1 IBD}) \mathbb{P}(\text{i & j 1 IBD}) + \mathbb{P}(\text{I & J IBD} |~ \text{i & j 2 IBD}) \mathbb{P}(\text{i & j 2 IBD}) \label{eqn:coeffkinship_step}\\[4pt] = & 0 \times r_0 + \frac{1}{4} r_1 + \frac{1}{2} r_2. \label{eqn:coeffkinship}\end{aligned}\]
In the above step, \ref{eqn:coeffkinship_step}, we’re summing the conditional probability of alleles \(I\) & \(J\) being IBD over whether our individuals \(i\) & \(j\) share \(0\) , \(1\) , or \(2\) alleles IBD, an example of using the Law of Total Probability (see Appendix \ref{eqn:law_tot_prob}). We’ve then, in \ref{eqn:coeffkinship}, used the fact that we can calculate our condition probabilities of I & J being IBD using the rules of Mendelian transmision. Consider the probability \(P(\text{i & j IBD} |~ \text{i & j 1 IBD})\) , i.e. that our pair of alleles ( \(I\) & \(J\) ) drawn from individuals \(i\) and \(j\) are IBD given that \(i\) and \(j\) share one allele IBD, this is a \(\frac{1}{4}\) as we need to draw the allele that is IBD from both \(i\) and \(j\) , i.e. drawing both black alleles in the middle panel of Figure \ref{fig:IBD_0_1_2}, which happens with probability \(\frac{1}{2} \times \frac{1}{2}\) . The coefficient of kinship will appear multiple times, in both our discussion of inbreeding and in the context of phenotypic resemblance between relatives.
Table \(\PageIndex{4}\): Probability that two individuals (i and j) of a given re- lationship share 0, 1, or 2 alleles identical by descent on the autosomes. ∗Assuming that our individuals are outbred and that this the only close relationship the pair shares.
| Relationship (i,j) \(^{*}\) | \(\mathbb{P}(\text{i & j 0 IBD})\) | \(\mathbb{P}(\text{i & j 1 IBD})\) | \(P(\text{i & j 2 IBD})\) | \(\mathbb{P}(\text{i & j IBD} )\) |
|---|---|---|---|---|
| Relationship (i,j) \(^{*}\) | \(r_0\) | \(r_1\) | \(r_2\) | \(F_{ij}\) |
| parent–child | 0 | 1 | 0 | |
| full siblings | ||||
| Monozygotic twins | 0 | 0 | 1 | |
| \(1^{st}\) cousins | 0 |
What are \(r_0\) , \(r_1\) , and \(r_2\) for \(\frac{1}{2}\) sibs? ( \(\frac{1}{2}\) sibs share one parent but not the other).
Explain in words why \(\mathbb{P}(\text{i & j IBD} |~ \text{i & j 2 IBD}) = \frac{1}{2}\) .
Genotypic sharing between pairs of individuals
Our \(r\) coefficients are going to have various uses. For example, they allow us to calculate the probability of the genotypes of a pair of relatives. Consider a biallelic locus where allele \(A_1\) is at frequency \(p\) , and two individuals who have IBD allele sharing probabilities \(r_0\) , \(r_1\) , \(r_2\) . What is the overall probability that these two individuals are both homozygous for allele 1? Well that’s
\[\begin{aligned} \mathbb{P}(\textrm{both } A_1 A_1) = & \mathbb{P}(\textrm{both } A_1 A_1 | 0 \text{ alleles } \mathrm{IBD}) \mathbb{P}(0 \text{ alleles } \mathrm{IBD}) + \mathbb{P}(\textrm{both } A_1 A_1 | 1 \text{ allele }IBD) \mathbb{P}(1\text{ allele }IBD) + \mathbb{P}(\textrm{both } A_1 A_1 | 2 \text{ alleles }IBD) \mathbb{P}(2\text{ alleles }IBD)\end{aligned}\]
Or, in our \(r_0\) , \(r_1\) , \(r_2\) notation:
\[\begin{aligned} \mathbb{P}(\textrm{both } A_1 A_1) = & \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{0 alleles IBD}) r_0 + \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{1 alleles IBD}) r_1 + \mathbb{P}(\textrm{both } A_1 A_1 | \textrm{2 alleles IBD}) r_2 \label{eqn:initial_relly_IBD_calc}\end{aligned}\]
If our pair of relatives share \(0\) alleles IBD, then the probability that they are both homozygous is \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{0 alleles IBD}) =p^2 \times p^2\) , as all four alleles represent independent draws from the population. If they share \(1\) allele IBD, then the shared allele is of type \(A_1\) with probability \(p\) , and then the other non-IBD allele, in both relatives, also needs to be \(A_1\) which happens with probability \(p^2\) , so \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{1 alleles IBD})=p \times p^2\) . Finally, our pair of relatives can share two alleles IBD, in which case \(\mathbb{P}(\textrm{both } A_1 A_1 | \text{2 alleles IBD}) = p^2\) , because if one of our individuals is homozygous for the \(A_1\) allele, both individuals will be. Putting this all together our \ref{eqn:initial_relly_IBD_calc} becomes
\[\mathbb{P}(\textrm{both } A_1 A_1) = p^4 r_0 + p^3 r_1 + p^2 r_2 \label{eqn:IBD_relly_calc}\]
Note that for specific cases we could also calculate this by summing over all the possible genotypes their shared ancestor(s) had; however, that would be much more involved and not as general as the form we have derived here.
We can write out terms like \ref{eqn:IBD_relly_calc} for all of the possible configurations of genotype sharing/non-sharing between a pair of individuals. Based on this we can write down the expected number of polymorphic sites where our individuals are observed to share 0, 1, or 2 alleles.
The genotype of our suspect in Question \ref{Q:CODIS} turns out to be 100/80 for D16S539 and 70/80 at TH01. The suspect is not a match to the DNA from the crime scene; however, they could be a sibling.Calculate the joint probability of observing the genotype from the crime and our suspect:
- Assuming that they share no close relationship.
- Assuming that they are full sibs.
- Briefly explain your findings.
There’s a variety of ways to estimate the relationships among individuals using genetic data. An example of using allele sharing to identify relatives is offered by the work of Nancy Chen . has collected genotyping data from thousands of Florida Scrub Jays at over ten thousand loci. These Jays live at the Archbold field site, and have been carefully monitored for many decades allowing the pedigree of many of the birds to be known. Using these data, she estimates allele frequencies at each locus. Then by equating the observed number of times that a pair of individuals share \(0\) , \(1\) , or \(2\) alleles to the theoretical expectation, she estimates the probability of \(r_0\) , \(r_1\) , and \(r_2\) for each pair of birds. A plot of these are shown in Figure \(\PageIndex{13}\), showing how well the estimates match those known from the pedigree.
Sharing of genomic blocks among relatives
We can more directly see the sharing of the genome among close relatives using high-density SNP genotyping arrays. In Figure \ref{fig:first_cousin_IBD} we show a simulation of first cousins’ genomic sharing from their shared grandmother. Colored purple are regions where they have matching genomic material, due to having inherited it IBD from their shared grandmother.
First cousins will share at least one allele of your genotype at all of the polymorphic loci in these purple regions. There’s a range of methods to detect such sharing. One way is to look for unusually long stretches of the genome where two individuals are never homozygous for different alleles. By identifying pairs of individuals who share an unusually large number of such putative IBD blocks, we can hope to identify unknown relatives in genotyping datasets. In fact, companies like 23&me and Ancestry.com use signals of IBD to help identify family ties.
As another example, consider the case of third cousins. You share one of eight sets of great-great-grandparents with each of your (likely many) third cousins. On average, you and each of your third cousins each inherit one-sixteenth of your genome from each of those two great-great-grandparents. This turns out to imply that on average, a little less than one percent of your and your third cousin’s genomes ( \(2 \times (1/16)^2 =0.78\%\) ) will be identical by virtue of descent from those shared ancestors. A simulated example where third cousins share blocks of their genome (on chromosome 16 and 2) due to their great-great-grandmother is shown in Figure \(\PageIndex{14}\).
Note how if you compare Figure \(\PageIndex{15}\) and Figure \(\PageIndex{14}\), individuals inherit less IBD from a shared great-great-grandmother than from a shared grandmother, as they inherit from more total ancestors further back. Also notice how the sharing occurs in shorter genomic blocks, as it has passed through more generations of recombination during meiosis. These blocks are still detectable, and so third cousins can be detected using high-density genotyping chips, allowing more distant relatives to be identified than single marker methods alone. More distant relations than third cousins, e.g. fourth cousins, start to have a significant probability of sharing none of their genome IBD. But you have many fourth cousins, so you will share some of your genome IBD with some of them; however, it gets increasingly hard to identify the degree of relatedness from genetic data the deeper in the family tree this sharing goes.
Inbreeding
We can define an inbred individual as an individual whose parents are more closely related to each other than two random individuals drawn from some reference population.
When two related individuals produce an offspring, that individual can receive two alleles that are identical by descent, i.e. they can be homozygous by by descent (sometimes termed autozygous), due to the fact that they have two copies of an allele through different paths through the pedigree. This increased likelihood of being homozygous relative to an outbred individual is the most obvious effect of inbreeding. It is also the one that will be of most interest to us, as it underlies a lot of our ideas about inbreeding depression and population structure. For example, in Figure \(\PageIndex{16}\) our offspring of first cousins is homozygous by descent having received the same IBD allele via two different routes around an inbreeding loop.
As the offspring receives a random allele from each parent ( \(i\) and \(j\) ), the probability that those two alleles are identical by descent is equal to the kinship coefficient \(F_{ij}\) of the two parents (\ref{eqn:coeffkinship}). This follows from the fact that the genotype of the offspring is made by sampling an allele at random from each of our parents.
| \(f_{11}\) | \(f_{12}\) | \(f_{22}\) |
|---|---|---|
| \((1-F) p^2 + F p\) | \((1-F) 2pq\) | \((1-F) q^2 + F q\) |
The only way the offspring can be heterozygous ( \(A_1 A_2\) ) is if their two alleles at a locus are not IBD (otherwise they would necessarily be homozygous). Therefore, the probability that they are heterozygous is
\[\mathbb{P}(A_1 A_2) = \mathbb{P}(A_1 A_2 |\textrm{I & J not IBD} ) \P (\textrm{I & J not IBD} ) = 2p q (1-F_{ij}) , \label{eq:hetGenHW}\]
The offspring can be homozygous for the \(A_1\) allele in two different ways. They can have two non-IBD alleles that are not IBD but happen to be of the allelic type \(A_1\) , or their two alleles can be IBD, such that they inherited allele \(A_1\) by two different routes from the same ancestor. Thus, the probability that an offspring is homozygous for \(A_1\) is
\[\begin{aligned} P(A_1 A_1) & = \mathbb{P}(A_1 A_1 |\textrm{I & J not IBD} ) \P (\textrm{i & j not IBD} ) + \mathbb{P}(A_1 A_1 |\textrm{i & j IBD} ) \P (\textrm{i & j IBD}) \nonumber\\ &= p^2(1-F_{ij}) + pF_{ij}.\end{aligned}\]
using the Law of Total Probability (see Appendix \ref{eqn:law_tot_prob}). Therefore, the frequencies of the three possible genotypes can be written as given in Table \(\PageIndex{5}\), which provides a generalization of the Hardy–Weinberg proportions.
The frequency of the \(A_1\) allele is \(p\) at a biallelic locus. Assume that our population is randomly mating and that the genotype frequencies in the population follow from HW. We select two individuals at random to mate from this population. We then mate the children from this cross. What is the probability that the child from this full sib-mating is homozygous?
Multiple inbreeding loops in a pedigree
Up to this point we have assumed that there is at most one inbreeding loop in the recent family history of our individuals, i.e. the parents of our inbred individual have at most one recent genealogical connection. However, an individual who has multiple inbreeding loops in their pedigree can be homozygous by descent thanks to receiving IBD alleles via multiple different different loops. To calculate inbreeding in pedigrees of arbitrary complexity, we can extend beyond our original relatedness coefficients \(r_0\) , \(r_1\) , and \(r_2\) to account for higher order sharing of alleles IBD among relatives. For example, we can ask, what is the probability that both of the alleles in the first individual are shared IBD with one allele in the second individual? There are nine possible relatedness coefficients in total to completely describe kinship between two diploid individuals, and we won’t go in to them here as it’s a lot to keep track of. However, we will show how we can calculate the inbreeding coefficient of an individual with multiple inbreeding loops more directly.
Let’s say the parents of our inbred individual (B and C) have \(K\) shared ancestors, i.e. individuals who appear in both B and C’s recent family trees. We denote these shared ancestors by \(A_1, \dots,A_K\) , and we denote by \(n\) the total number of individuals in the chain from B to C via ancestor \(A_i\) , including B, C, and \(A_i\) . For example, if B is C’s aunt, then B and C share two ancestors, which are B’s parents and, equivalently, C’s grandparents. In this case, there are n=4 individuals from B to C through each of these two shared ancestor. In the general case, the kinship coefficient of B and C, i.e. the inbreeding coefficient of their child, is
\[F = \sum_{i=1}^K \frac{1}{2^{n_i}} \big( 1+ f_{A_i} \big) \label{eqn:inbreeding_over_ancs}\]
where \(f_{A_i}\) is the inbreeding coefficient of the ancestor \(A_i\) . What’s happening here is that we sum over all the mutually-exclusive paths in the pedigree through which B and C can share an allele IBD. With probability \(\frac{1}{2^{n_i}}\) , a pair of alleles picked at random from B and C is descended from the same ancestral allele in individual \(A_i\) , in which case the alleles are IBD. However, even if B inherits the maternal allele and C inherits the paternal allele of shared ancestor \(A_i\) , if \(A_i\) was themselves inbred, with probability \(f_{A_i}\) those two alleles are themselves IBD. Thus a shared inbred ancestor further increases the kinship of B and C.
Multiple inbreeding loops increase the probability that a child is homozygous by descent at a locus, which can be calculated simply by plugging in \(F\) , the child’s inbreeding coefficient, into our generalized HW equation.
As one extreme example of the impact of multiple inbreeding loops in an individual’s pedigree, let’s consider king Charles II of Spain, the last of the Spanish Habsburgs. Charles was the son of Philip IV of Spain and Mariana of Austria, who were uncle and niece. If this were the only inbreeding loop, then Charles would have had an inbreeding coefficient of \(\frac{1}{8}\) . Unfortunately for Charles, the Spanish Habsburgs had long kept wealth and power within their family by arranging marriages between close kin. The pedigree of Charles II is shown in Figure \(\PageIndex{17}\), and multiple inbreeding loops are apparent. For example, Phillip III, Charles II’s grandfather and great-grandfather, was himself a child of an uncle-niece marriage.
calculated that Charles II had an inbreeding coefficient of \(0.254\) , equivalent to a full-sib mating, thanks to all of the inbreeding loops in his pedigree. Therefore, he is expected to have been homozygous by descent for a full quarter of his genome. As we’ll talk about later in these notes, this means that Charles may have been homozygous for a number of recessive disease alleles, and indeed he was a very sickly man who left no descendants due to his infertility. Thus plausibly the end of one of the great European dynasties came about through inbreeding.
Calculating inbreeding coefficients from genetic data
If the observed heterozygosity in a population is \(H_O\) , and we assume that the generalized Hardy–Weinberg proportions hold, we can set \(H_O\) equal to \(f_{12}\) , and solve Equation \ref{eq:hetGenHW} for \(F\) to obtain an estimate of the inbreeding coefficient as
\[\hat{F} = 1-\frac{f_{12}}{2pq} = \frac{2pq - f_{12}}{2pq}. \label{eqn:Fhat}\]
As before, \(p\) is the frequency of allele \(A_{1}\) in the population. This can be rewritten in terms of the observed heterozygosity ( \(H_O\) ) and the heterozygosity expected in the absence of inbreeding, \(H_E=2pq\) , as
\[\hat{F} = \frac{H_E-H_O}{H_E} = 1 - \frac{H_O}{H_E}. \label{eqn:FhatHO}\]
Hence, \(\hat{F}\) quantifies the deviation due to inbreeding of the observed heterozygosity from the one expected under random mating, relative to the latter.
Suppose the following genotype frequencies were observed for an esterase locus in a population of Drosophila (A denotes the “fast" allele and B denotes the “slow" allele):
| AA | AB | BB |
|---|---|---|
| 0.6 | 0.2 | 0.2 |
What is the estimate of the inbreeding coefficient at the esterase locus?
If we have multiple loci, we can replace \(H_O\) and \(H_E\) by their means over loci, \(\bar{H}_O\) and \(\bar{H}_E\) , respectively. Note that, in principle, we could also calculate \(F\) for each individual locus first, and then take the average across loci. However, this procedure is more prone to introducing a bias if sample sizes vary across loci, which is not unlikely when we are dealing with real data.
Genetic markers are commonly used to estimate inbreeding for wild and/or captive populations of conservation concern. As an example of this, consider the case of the Mexican wolf ( Canis lupus baileyi ), a sub-species of gray wolf.
They were extirpated in the wild during the mid-1900s due to hunting, and the remaining five Mexican wolves in the wild were captured to start a breeding program. estimated the current-day, average expected heterozygosity to be \(0.18\) , based on allele frequencies at over forty thousand SNPs. However, the average Mexican wolf individual was only observed to be heterozygous at \(12\%\) of these SNPs. Therefore, the average inbreeding coefficient for the Mexican wolf is \(\hat{F} = 1 -\frac{0.12}{0.18}\) , i.e. \(\sim 33 \%\) of a lobo’s genome is homozygous due to recent inbreeding in their pedigree.
Genomic blocks of homozygosity due to inbreeding.
As we saw above, close relatives are expected to share alleles IBD in large genomic blocks. Thus, when related individuals mate and transmit alleles to an inbred offspring, they transmit these alleles in big blocks through meiosis. As an example, let’s return to the case of our hypothetical first cousins from Figure \ref{fig:IBD_cousins_chr_cartoon}. If this pair of individuals had a child, one possible pattern of genetic transmission is shown in Figure \(\PageIndex{20}\). The child has inherited the red stretch of chromosome via two different routes through their predigree from the grandparents. This is an example of an autozygous segment, where the child is homozygous by descent at all of the loci in this red region.
The inbreeding coefficient of the child sets the proportion of their genome that will be in these autozygous segments. For example, a child of first full cousins is expected to have \(1/16\) of their genome in these segments. The more distant the loop in the pedigree, the more meioses that chromosomes have been through and the shorter individual blocks will be. A child of first cousins will have longer blocks than a child of second cousins, for example.
Individuals with multiple inbreeding loops in their family tree can have a high inbreeding coefficient due to the combined effect of many small blocks of autozygosity. For example, Charles II had an inbreeding coefficient that is equivalent to that of the child of full-sibs, with a quarter of his genome expected to homozygous by descent, but this would be made up of many shorter blocks.
We can hope to detect these blocks by looking for unusually long genomic runs of homozygosity (ROH) sites in an individual’s genome. One way to estimate an individual’s inbreeding coefficient is then to total up the proportion of an individual’s genome that falls in such ROH regions. This estimate is called \(F_{ROH}\) .
An example of using \(F_{ROH}\) to study inbreeding comes from the work of Sams and Boyko (2018b), who identified runs of homozygosity in 2,500 dogs, ranging from 500kb up to many megabases. Figure \(\PageIndex{21}\) shows the distribution of FROH of individuals in each dog breed for the X and autosome. In Figure \(\PageIndex{23}\) this is broken down by the length of ROH segments.
Dog breeds have been subject to intense breeding that has resulted in high levels of inbreeding. Of the population samples examined, Doberman Pinschers have the highest levels of their genome in runs of homozygosity ( \(F_{ROH}\) ), somewhat higher than English bulldogs. In Figure \(\PageIndex{23}\) we can see that English bulldogs have more short ROH than Doberman Pinschers, but that Doberman Pinschers have more of their genome in very large ROH ( \(>16 Mb\) ). This suggests that English bulldogs have had long history of inbreeding as they have many small blocks, but that Doberman Pinschers have a lot of recent inbreeding as their autozygosity is contained in long blocks relatively unbroken by recombination.
Summary
- This chapter developed the relationship between allele frequencies and genotype frequencies within a generation and among relatives.
- Under random mating, we derived expectations of the genotype frequencies (Hardy-Weinberg), and we can identify deviations away from these expectations.
- Identity by descent (IBD) refers to the sharing of alleles due to a recent shared biological relationship.
- We can predict the probability and expected level of sharing of alleles IBD among pairs of relatives using mendelian transmission probabilities (as contained in coefficients \(r_0\) , \(r_1\) , and \(r_2\) ). One useful summary of relatedness for a pair of individuals is the kinship coefficient \(F_{i,j}\) .
- We can also learn about genetic relationships from the sharing of genomic segments among relatives, with many long shared segments revealing a closer relationship.
- An inbred individual has parents who are more closely related than random draws from some reference population.
- Inbreeding results in decreased heterozygosity and a complementary increase in homozygosity. We can use the kinship coefficient of the parents to estimate the distortion away from Hardy-Weinberg and the expected level of heterozygosity.
- Inbreeding coefficients can be calculated from genetic data, either for multiple individuals at a single locus or for multiple loci for a single individual.
Calculate \(r_{0}\) , \(r_{1}\) , \(r_{2}\) and the coefficient of kinship \(F\) between:
- A grandparent and their grandchild
- A great grandparent and their great grandchild
- Full siblings
- A great aunt and her grand nephew (your great aunt = your parent’s aunt)
You are studying a codominant flower colour polymorphism. Skipping through a meadow of flowers you and compile the following data:
| red | pink | white |
| 200 | 100 | 200 |
- What frequencies would you expect at this locus under Hardy-Weinberg equilibrium?
- Calculate the inbreeding coefficient at this locus.
- Name two distinct processes that could lead to the deviation you see, and describe how they would result in a deficit of heterozygotes.
What are the relatedness coefficients of the X chromosome between:
- Two male full siblings?
- Two female full siblings?
- What is the probability that a female offspring of a full sib mating is homozygous by descent at a locus on her X chromosome?
You are studying the wing spot polymorphism in a butterfly species. From crosses in the lab you find that the presence of wing spots is determined by a dominant allele.
You collect 100 butterflies, 84 of them have the wing spots. What is the frequency of the wing-spot allele? What assumption did you have to make to come to your answer?
An allele has frequency of \(0.001\) in the population. What is the probability that both you and your first (full) cousin are heterozygote for the allele?
The kinship coefficient of the parents is the inbreeding coefficient of the offspring. Explain, with reference to the weighting of relatedness coefficients in the inbreeding coefficient, why the inbreeding coefficient is the probability that a locus is homozygous by descent.
In terms of identity by descent, explain why multiple inbreeding loops in an individual’s pedigree lead to higher levels of inbreeding.


40.2: Components of the Blood
Skills to Develop
- List the basic components of the blood
- Compare red and white blood cells
- Describe blood plasma and serum
Hemoglobin is responsible for distributing oxygen, and to a lesser extent, carbon dioxide, throughout the circulatory systems of humans, vertebrates, and many invertebrates. The blood is more than the proteins, though. Blood is actually a term used to describe the liquid that moves through the vessels and includes plasma (the liquid portion, which contains water, proteins, salts, lipids, and glucose) and the cells (red and white cells) and cell fragments called platelets . Blood plasma is actually the dominant component of blood and contains the water, proteins, electrolytes, lipids, and glucose. The cells are responsible for carrying the gases (red cells) and immune the response (white). The platelets are responsible for blood clotting. Interstitial fluid that surrounds cells is separate from the blood, but in hemolymph, they are combined. In humans, cellular components make up approximately 45 percent of the blood and the liquid plasma 55 percent. Blood is 20 percent of a person’s extracellular fluid and eight percent of weight.
The Role of Blood in the Body
Blood, like the human blood illustrated in Figure \(\PageIndex{1}\) is important for regulation of the body’s systems and homeostasis. Blood helps maintain homeostasis by stabilizing pH, temperature, osmotic pressure, and by eliminating excess heat. Blood supports growth by distributing nutrients and hormones, and by removing waste. Blood plays a protective role by transporting clotting factors and platelets to prevent blood loss and transporting the disease-fighting agents or white blood cells to sites of infection.
Red Blood Cells
Red blood cells , or erythrocytes (erythro- = “red”; -cyte = “cell”), are specialized cells that circulate through the body delivering oxygen to cells; they are formed from stem cells in the bone marrow. In mammals, red blood cells are small biconcave cells that at maturity do not contain a nucleus or mitochondria and are only 7–8 µm in size. In birds and non-avian reptiles, a nucleus is still maintained in red blood cells.
The red coloring of blood comes from the iron-containing protein hemoglobin, illustrated in Figure \(\PageIndex{2}\)a. The principal job of this protein is to carry oxygen, but it also transports carbon dioxide as well. Hemoglobin is packed into red blood cells at a rate of about 250 million molecules of hemoglobin per cell. Each hemoglobin molecule binds four oxygen molecules so that each red blood cell carries one billion molecules of oxygen. There are approximately 25 trillion red blood cells in the five liters of blood in the human body, which could carry up to 25 sextillion (25 × 10 21 ) molecules of oxygen in the body at any time. In mammals, the lack of organelles in erythrocytes leaves more room for the hemoglobin molecules, and the lack of mitochondria also prevents use of the oxygen for metabolic respiration. Only mammals have anucleated red blood cells, and some mammals (camels, for instance) even have nucleated red blood cells. The advantage of nucleated red blood cells is that these cells can undergo mitosis. Anucleated red blood cells metabolize anaerobically (without oxygen), making use of a primitive metabolic pathway to produce ATP and increase the efficiency of oxygen transport.
Not all organisms use hemoglobin as the method of oxygen transport. Invertebrates that utilize hemolymph rather than blood use different pigments to bind to the oxygen. These pigments use copper or iron to the oxygen. Invertebrates have a variety of other respiratory pigments. Hemocyanin, a blue-green, copper-containing protein, illustrated in Figure \(\PageIndex{2}\)b is found in mollusks, crustaceans, and some of the arthropods. Chlorocruorin, a green-colored, iron-containing pigment is found in four families of polychaete tubeworms. Hemerythrin, a red, iron-containing protein is found in some polychaete worms and annelids and is illustrated in Figure \(\PageIndex{2}\)c. Despite the name, hemerythrin does not contain a heme group and its oxygen-carrying capacity is poor compared to hemoglobin.
The small size and large surface area of red blood cells allows for rapid diffusion of oxygen and carbon dioxide across the plasma membrane. In the lungs, carbon dioxide is released and oxygen is taken in by the blood. In the tissues, oxygen is released from the blood and carbon dioxide is bound for transport back to the lungs. Studies have found that hemoglobin also binds nitrous oxide (NO). NO is a vasodilator that relaxes the blood vessels and capillaries and may help with gas exchange and the passage of red blood cells through narrow vessels. Nitroglycerin, a heart medication for angina and heart attacks, is converted to NO to help relax the blood vessels and increase oxygen flow through the body.
A characteristic of red blood cells is their glycolipid and glycoprotein coating; these are lipids and proteins that have carbohydrate molecules attached. In humans, the surface glycoproteins and glycolipids on red blood cells vary between individuals, producing the different blood types, such as A, B, and O. Red blood cells have an average life span of 120 days, at which time they are broken down and recycled in the liver and spleen by phagocytic macrophages, a type of white blood cell.
White Blood Cells
White blood cells, also called leukocytes (leuko = white), make up approximately one percent by volume of the cells in blood. The role of white blood cells is very different than that of red blood cells: they are primarily involved in the immune response to identify and target pathogens, such as invading bacteria, viruses, and other foreign organisms. White blood cells are formed continually; some only live for hours or days, but some live for years.
The morphology of white blood cells differs significantly from red blood cells. They have nuclei and do not contain hemoglobin. The different types of white blood cells are identified by their microscopic appearance after histologic staining, and each has a different specialized function. The two main groups, both illustrated in Figure \(\PageIndex{3}\) are the granulocytes, which include the neutrophils, eosinophils, and basophils, and the agranulocytes, which include the monocytes and lymphocytes.
Granulocytes contain granules in their cytoplasm; the agranulocytes are so named because of the lack of granules in their cytoplasm. Some leukocytes become macrophages that either stay at the same site or move through the blood stream and gather at sites of infection or inflammation where they are attracted by chemical signals from foreign particles and damaged cells. Lymphocytes are the primary cells of the immune system and include B cells, T cells, and natural killer cells. B cells destroy bacteria and inactivate their toxins. They also produce antibodies. T cells attack viruses, fungi, some bacteria, transplanted cells, and cancer cells.  Natural killer cells attack a variety of infectious microbes and certain tumor cells.
One reason that HIV poses significant management challenges is because the virus directly targets T cells by gaining entry through a receptor. Once inside the cell, HIV then multiplies using the T cell’s own genetic machinery. After the HIV virus replicates, it is transmitted directly from the infected T cell to macrophages. The presence of HIV can remain unrecognized for an extensive period of time before full disease symptoms develop.
Platelets and Coagulation Factors
Blood must clot to heal wounds and prevent excess blood loss. Small cell fragments called platelets (thrombocytes) are attracted to the wound site where they adhere by extending many projections and releasing their contents. These contents activate other platelets and also interact with other coagulation factors, which convert fibrinogen, a water-soluble protein present in blood serum into fibrin (a non-water soluble protein), causing the blood to clot. Many of the clotting factors require vitamin K to work, and vitamin K deficiency can lead to problems with blood clotting. Many platelets converge and stick together at the wound site forming a platelet plug (also called a fibrin clot), as illustrated in Figure \(\PageIndex{4}\)b. The plug or clot lasts for a number of days and stops the loss of blood. Platelets are formed from the disintegration of larger cells called megakaryocytes, like that shown in Figure \(\PageIndex{4}\)a. For each megakaryocyte, 2000–3000 platelets are formed with 150,000 to 400,000 platelets present in each cubic millimeter of blood. Each platelet is disc shaped and 2–4 μm in diameter. They contain many small vesicles but do not contain a nucleus.
Plasma and Serum
The liquid component of blood is called plasma, and it is separated by spinning or centrifuging the blood at high rotations (3000 rpm or higher). The blood cells and platelets are separated by centrifugal forces to the bottom of a specimen tube. The upper liquid layer, the plasma, consists of 90 percent water along with various substances required for maintaining the body’s pH, osmotic load, and for protecting the body. The plasma also contains the coagulation factors and antibodies.
The plasma component of blood without the coagulation factors is called the serum . Serum is similar to interstitial fluid in which the correct composition of key ions acting as electrolytes is essential for normal functioning of muscles and nerves. Other components in the serum include proteins that assist with maintaining pH and osmotic balance while giving viscosity to the blood. The serum also contains antibodies, specialized proteins that are important for defense against viruses and bacteria. Lipids, including cholesterol, are also transported in the serum, along with various other substances including nutrients, hormones, metabolic waste, plus external substances, such as, drugs, viruses, and bacteria.
Human serum albumin is the most abundant protein in human blood plasma and is synthesized in the liver. Albumin, which constitutes about half of the blood serum protein, transports hormones and fatty acids, buffers pH, and maintains osmotic pressures. Immunoglobin is a protein antibody produced in the mucosal lining and plays an important role in antibody mediated immunity.
Evolution Connection: Blood Types
Related to Proteins on the Surface of the Red Blood Cells Red blood cells are coated in antigens made of glycolipids and glycoproteins. The composition of these molecules is determined by genetics, which have evolved over time. In humans, the different surface antigens are grouped into 24 different blood groups with more than 100 different antigens on each red blood cell. The two most well known blood groups are the ABO, shown in Figure \(\PageIndex{5}\), and Rh systems. The surface antigens in the ABO blood group are glycolipids, called antigen A and antigen B. People with blood type A have antigen A, those with blood type B have antigen B, those with blood type AB have both antigens, and people with blood type O have neither antigen. Antibodies called agglutinougens are found in the blood plasma and react with the A or B antigens, if the two are mixed. When type A and type B blood are combined, agglutination (clumping) of the blood occurs because of antibodies in the plasma that bind with the opposing antigen; this causes clots that coagulate in the kidney causing kidney failure. Type O blood has neither A or B antigens, and therefore, type O blood can be given to all blood types. Type O negative blood is the universal donor. Type AB positive blood is the universal acceptor because it has both A and B antigen. The ABO blood groups were discovered in 1900 and 1901 by Karl Landsteiner at the University of Vienna.
The Rh blood group was first discovered in Rhesus monkeys. Most people have the Rh antigen (Rh+) and do not have anti-Rh antibodies in their blood. The few people who do not have the Rh antigen and are Rh– can develop anti-Rh antibodies if exposed to Rh+ blood. This can happen after a blood transfusion or after an Rh– woman has an Rh+ baby. The first exposure does not usually cause a reaction; however, at the second exposure, enough antibodies have built up in the blood to produce a reaction that causes agglutination and breakdown of red blood cells. An injection can prevent this reaction.
Summary
Specific components of the blood include red blood cells, white blood cells, platelets, and the plasma, which contains coagulation factors and serum. Blood is important for regulation of the body’s pH, temperature, osmotic pressure, the circulation of nutrients and removal of waste, the distribution of hormones from endocrine glands, and the elimination of excess heat; it also contains components for blood clotting. Red blood cells are specialized cells that contain hemoglobin and circulate through the body delivering oxygen to cells. White blood cells are involved in the immune response to identify and target invading bacteria, viruses, and other foreign organisms; they also recycle waste components, such as old red blood cells. Platelets and blood clotting factors cause the change of the soluble protein fibrinogen to the insoluble protein fibrin at a wound site forming a plug. Plasma consists of 90 percent water along with various substances, such as coagulation factors and antibodies. The serum is the plasma component of the blood without the coagulation factors.
Glossary
- plasma
- liquid component of blood that is left after the cells are removed
- platelet
- (also, thrombocyte) small cellular fragment that collects at wounds, cross-reacts with clotting factors, and forms a plug to prevent blood loss
- red blood cell
- small (7–8 μm) biconcave cell without mitochondria (and in mammals without nuclei) that is packed with hemoglobin, giving the cell its red color; transports oxygen through the body
- serum
- plasma without the coagulation factors
- white blood cell
- large (30 μm) cell with nuclei of which there are many types with different roles including the protection of the body from viruses and bacteria, and cleaning up dead cells and other waste


6.10: Exercises
1. We say that a relation \(R\) on a set \(X\) is symmetric if \((x,y) \in R\) implies \((y,x) \in R\) for all \(x,y \in X\). If \(X=\{a,b,c,d,e,f\}\), how many symmetric relations are there on \(X\)? How many of these are reflexive?
2. A relation \(R\) on a set \(X\) is an equivalence relation if \(R\) is reflexive, symmetric, and transitive. Fix an integer \(m \geq 2\). Show that the relation defined on the set \(\mathbb{Z}\) of integers by \(aRb (a,b \in \mathbb{Z})\) if and only if \(a≡b(mod m)\) is an equivalence relation. (Recall that \(a≡b(mod m\)) means that when dividing \(a\) by \(m\) and \(b\) by \(m\) you get the same remainder.)
\(P=\{(1,1),(2,2),(3,3),(4,4),(1,3),(2,4),(2,5),(4,5),(3,5),(1,5)\}\)
a partial order on the set \(X=\{1,2,3,4,5\}\)? If so, discuss what properties you verified and how. If not, list the ordered pairs that must be added to \(P\) to make it a partial order or say why it cannot be made a partial order by adding ordered pairs.
4. Draw the diagram of the poset \(\textbf{P}=(X,P)\) where \(X=\{1,2,3,5,6,10,15,30\}\) and \(x \leq y\) in \(P\) if and only if \(x|y\). (Recall that \(x|y\) means that \(x\) evenly divides \(y\) without remainder. Equivalently \(x|y\), if and only if \(y≡0\) (mod \(x\)).)
5. Draw the diagram of the poset \(\textbf{P}=(X,P)\) where
\(X=\{\{1,3,4,5,6\},\{1,2,4,5,6\},\{1,2,3,6\},\{1,2,3\},\{1,5,6\},\{1,3,6\},\{1,2\},\{1,6\},\{3,5\},\{1\},\{3\},\{4\}\}\)
and \(P\) is the partial order on \(X\) given by the “is a subset of” relationship.
6. A linear extension of a poset \(\textbf{P}=(X,P)\) is a total order \(L\) on \(X\) such that if \(x \leq y\) in \(P\), then \(x \leq y\) in \(L\). Give linear extension of the three posets shown in Figure 6.8 . If you feel very ambitious, try to count the number of linear extensions of the poset on the left side of the figure. Don't list them. Just provide an integer as your answer.
7. Alice and Bob are considering posets \(\textbf{P}\) and \(\textbf{Q}\). They soon realize that \(\textbf{Q}\) is isomorphic to \(\textbf{P}^d\). After 10 minutes of work, they figure out that \(\textbf{P}\) has height 5 and width 3. Bob doesn't want do find the height and width of \(\textbf{Q}\), since he figures it will take (at least) another 10 minutes to answer these questions for \(\textbf{Q}\). Alice says Bob is crazy and that she already knows the height and width of \(\textbf{Q}\). Who's right and why?
8. For this exercise, consider the poset \(\textbf{P}\) in Figure 6.5 .
a. List the maximal elements of \(\textbf{P}\).
b. List the minimal elements of \(\textbf{P}\).
c. Find a maximal chain with two points in \(\textbf{P}\).
d. Find a chain in \(\textbf{P}\) with three points that is not maximal. Say why your chain is not maximal.
e. Find a maximal antichain with four points in \(\textbf{P}\).
9. Find the height \(h\) of the poset \(\textbf{P}=(X,P)\) shown below as well as a maximum chain and a partition of \(X\) into \(h\) antichains using the algorithm from this chapter.
11. A restaurant chef has designed a new set of dishes for his menu. His set of dishes contains 10 main courses, and he will select a subset of them to place on the menu each night. To ensure variety of main courses for his patrons, he wants to guarantee that a night's menu is neither completely contained in nor completely contains another night's menu. What is the largest number of menus he can plan using his 10 main courses subject to this requirement?
12. Draw the diagram of the interval order represented in Figure 6.34 .
13. Draw the diagram of the interval order represented in Figure 6.35.
14. Find an interval representation for the poset in Figure 6.36 or give a reason why one does not exist.
15. Find an interval representation for the poset in Figure 6.37 or give a reason why one does not exist.
16. Find an interval representation for the poset in Figure 6.38 or give a reason why one does not exist.
17. Find an interval representation for the poset in Figure 6.39 or give a reason why one does not exist.
18. Use the First Fit algorithm (ordering by left endpoints) to find the width \(w\) of the interval order shown in Figure 6.40 and a partition into \(w\) chains. Also give an antichain with \(w\) points.
19. Complete the proof of Theorem 6.30 .
- Hint
-
The key idea is to show that if \(d\) is the least positive integer for which an interval order \(\textbf{P}\) has a representation using end points from \(\{1,2,…,n\}\), then every integer \(i\) from this set must be both a left end point and a right end point of an interval.
20. Show that every poset is isomorphic to a poset of each of the four types illustrated in Example 6.7 .
- Hint
-
For each element \(x\), choose some unique identifying key which is an element/prime/coordinate/observer. Then associate with \(x\) a structure that identifies the keys of elements from \(D[x]\).
21. The dimension of a poset \(\textbf{P}=(X,P)\), denoted dim(\(\textbf{P}\)), is the least \(t\) for which \(P\) is the intersection of \(t\) linear orders on \(X\).
a. Show that the dimension of a poset \(\textbf{P}\) is the same as the dimension of its dual.
b. Show that \(\textbf{P}\) is a subposet of \(\textbf{Q}\), then dim(\(\textbf{P}\))≤dim(\(\textbf{Q}\)).
c. Show that the removal of a point can reduce the dimension by at most 1.
d. Find the dimension of the posets in Figure 6.8 .
e. Use Dilworth's theorem to show that the dimension of a poset is at most its width.
f. Use the example on the left side of Figure 6.33 to show that for every \(n \geq 2\), there exists a poset \(\textbf{P}_n\) on \(2n\) points having width and dimension equal to \(n\).


5.1.3F: Groups
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Groups are collections of people who identify and interact with one another and are united in some way.
Learning Objectives
- Discuss how heuristics allow people to learn about people within a society based on group affiliation and give examples of both positive and negative heuristics
A social group is a collection of people who identify and interact with one another. They are united in some way, meaning that they may share interests, values, language, backgrounds, social roles, or family ties.
Perceiving society as a composite of groups allows for one to generalize about a particular person. If you know something about which groups a particular person belongs to, you may import certain character traits to that particular individual. In this way, groups operate as a cognitive heuristic, meaning that people use groups as a shortcut to use generalized information to learn about a particular person. If you know that Johnny is a member of the Computer Geek Club, you will assume that Johnny likes computers. Thus, you use group identity to generalize and make assumptions about a specific individual. Heuristics are sometimes perceived to be legitimate assumptions about an individual and sometimes deemed illegitimate. Legitimate heuristics tend to just be those that import positive generalizations to a particular person. For example, if you know that Sue holds an undergraduate degree, you may assume that Sue is an educated individual. However, the same heuristic can function in negative ways; this is the underlying mechanism that enables stereotypes. Stereotyping is when one makes generalizations about a particular person in a negative way based on their perceived group identity. Racism is one example of this; making assumptions about an individual because of their racial background is an example of negative group heuristics and stereotyping. Both legitimate and illegitimate heuristics demonstrate how knowledge about one’s group affiliations conveys perceived social knowledge about that individual.
In this way, a person’s group affiliation is associated with that person’s identity. The relationship between identity and politics is dynamic and contested. Sometimes a person is perceived to be part of a group to which they identify, such as when someone makes assumptions about another person’s identity based on their racial background. Conversely, an individual may claim membership to a group that rejects the petition. Take, for example, Elizabeth Warren’s 2012 campaign for the United States Senate. In public remarks, Warren declared that she had Native American heritage and was part Cherokee. However, many Cherokee individuals protested the declaration and insisted that Ms. Warren was not Cherokee until she could prove otherwise. By this, one can see how groups can serve as an interface between an individual and society at large.
Key Points
- Groups can be united by shared interests, values, language, backgrounds, social roles, or family ties.
- Group identification is one way that individuals make assumptions about others’ identities.
- Groups operate as cognitive heuristics. Heuristics are sometimes perceived to be legitimate assumptions about an individual and sometimes deemed illegitimate. Legitimate heuristics tend to just be those that import positive generalizations to a particular person.
- Group affiliation is negotiated. Sometimes an individual will protest an assumed group affiliation. Other times, a group will contest an individual’s claim to membership.
Key Terms
- Legitimate Heuristic : Generally, a heuristic that imports positive generalizations to a particular person.
- stereotype : A conventional, formulaic, and oversimplified conception, opinion, or image of a group of people or things.
- heuristic : An experience-based technique for problem solving, learning, and discovery. Examples include using a rule of thumb or making an educated guess.


7.11: Play- The Vehicle for Development and Learning
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Since in previous chapters we have explored the notion that children learn through “play”, let’s expand on that concept a bit as it relates to curriculum.
Children are born observers and are active participants in their own learning and understanding of the world around them from the very beginning of their existence. This means they are not just recipients of a teacher’s knowledge. Developmentally Appropriate Practice (DAP) challenges early childhood professionals to be intentional in their interactions and environments to create optimal experiences to maximize children’s growth and development. Under this umbrella of DAP, knowledge is based upon discovery and discovery occurs through active learning and abundant opportunities for exploration. Through a “hands-on” approach and using play as a vehicle, children will develop the skills necessary for growth and development and maximize their learning.
Teachers play a pivotal role in children’s active construction of knowledge. They intentionally provide the environments, interactions, and experiences that support children in actively building concepts, skills, and overall development. The role of the teacher who works with young children in early childhood is to support children’s active construction of knowledge. In a sense, early childhood teachers serve as research supports as the children sense, discover, and construct meaning about the world around them.
Early childhood teachers are responsible for:
- offering children well-stocked play spaces where they can construct concepts and ideas, preferably in the company of peers
- designing daily routines that invite children to be active participants and to use emerging skills and concepts
- supporting children’s learning through interactions and conversations that prompt using language and ideas in new ways
Pause to Reflect
Going back to Table 6.1, which concepts are being developed by which types of play? By incorporating each of these types of play, are you developing the “whole child”? Why or why not?
As with most things, the way that children play will go through developmental stages. As teachers plan, they keep in mind the stages of play relevant to the children they are planning for. Originally described by Parten (1932), this list, explains how children’s play changes by age as they grow and develop social skills.
- Unoccupied Play (Birth-3 Months) : At this stage, a baby is making many movements with their arms, legs, hands, feet, etc. They are learning about and discovering how their body moves.
- Solitary Play (Birth-2 Years) : This is the stage when a child plays alone. They are not interested in playing with others quite yet.
- Spectator/Onlooker Behavior (2 Years) : During this stage, a child begins to watch other children playing but does not play with them.
- Parallel Play (2+Years) : When a child plays alongside or near others but does not play with them.
- Associate Play (3-4 Years) : When a child starts to interact with others during play, but there is not a large amount of interaction at this stage.
- Cooperative Play (4+ years) : When a child plays together with others and has interest in both the activity and other children involved in playing. [73]
Pause to Reflect
Why might these stages be important to consider? How would you use this information in your planning?


19.2: Anthropogenic Climate Change
When we talk about anthropogenic climate change, we are generally thinking of the industrial era, which really got going when we started using fossil fuels (coal to begin with) to drive machinery and trains. That was around the middle of the 18th century. The issue with fossil fuels is that they involve burning carbon that was naturally stored in the crust over hundreds of millions of years as part of Earth’s process of counteracting the warming Sun.
Some climate scientists argue that anthropogenic climate change actually goes back much further than the industrial era, and that humans began to impact the climate by clearing land to grow grains in Europe and the Middle East around 8,000 years BCE and by creating wetlands to grow rice in Asia around 5,000 years BCE. Clearing forests for crops is a type of climate-forcing because the CO 2 storage capacity of the crops is generally lower than that of the trees they replace, and creating wetlands is a type of climate forcing because the anaerobic bacterial decay of organic matter within wetlands produces CH 4 .
In fact, whether anthropogenic climate change started with the agricultural revolution or the industrial revolution is not important, because the really significant climate changes didn’t start until the early part of the 20th century, and although our activities are a major part of the problem, our increasing numbers are a big issue as well. Figure \(\PageIndex{1}\) shows the growth of the world population from around 5 million, when we first started growing crops, to about 18 million when wetland rice cultivation began, to over 800 million at the start of the industrial revolution, to over 7,700 million in 2019. A big part of the incredible growth in our population is related to the availability of the cheap and abundant energy embodied in fossil fuels, which we use for transportation, heating and cooling, industry, and food production. It will be hard to support a population of this size without fossil fuels, but we have to find a way to do it.
A rapidly rising population, the escalating level of industrialization and mechanization of our lives, and an increasing dependence on fossil fuels for transportation and energy generation have driven the anthropogenic climate change of the past century. The trend of mean global temperatures since 1880 is shown in Figure \(\PageIndex{2}\). For approximately the past 55 years, the temperature has increased at a relatively steady and disturbingly rapid rate, especially compared to past changes. The average temperature now is approximately 1.0°C higher than before industrialization, and two-thirds of this warming has occurred since 1975. One of the driving factors of the recent increase in the rate of climate change has been the migration of North Americans from city centres to the suburbs, and the resulting need for virtually every household to own at least one car, when previously they were able to get around on foot or public transit.
The Intergovernmental Panel on Climate Change (IPCC)—established by the United Nations in 1988—is responsible for reviewing the scientific literature on climate change and issuing periodic reports on several topics, including the scientific basis for understanding climate change, our vulnerability to observed and predicted climate changes, and what we can do to limit climate change and minimize its impacts. Figure \(\PageIndex{3}\), which is based on data from the fifth assessment report of the IPCC, issued in 2014, shows the relative contributions of various long-lasting anthropogenic GHGs to current climate forcing, based on the changes from levels that existed in 1750.
The biggest anthropogenic contributor to warming is CO 2 , which accounts for 56% of positive forcing. CH 4 accounts for 32%, and the halocarbon gases (mostly leaked from older air-conditioning appliances that still contain CFCs) and nitrous oxide (N 2 O) (from burning fossils fuels) account for 6% each.
CO 2 emissions come mostly from coal- and gas-fired power stations, motorized vehicles (cars, trucks, and aircraft), and industrial operations (e.g., smelting). CH 4 emissions come from production of fossil fuels (escape from coal mining and from gas and oil production and processing), livestock farming (mostly beef), landfills, waste water, and wetland rice farming. N 2 O is derived almost entirely from the combustion of fossil fuels. In summary, most (by far) of our current GHG emissions come from fossil fuel production and use.
Figure \(\PageIndex{3}\) shows the proportions of climate forcing related to the 4 main greenhouse gases, and also includes lists of where those gases come from. The ones that are primarily derived from our use of fossil fuels are shown in red text.
Using the numbers provided in Figure \(\PageIndex{3}\) determine the percentage proportions of GHGs from the 4 main sources listed in the table below. Make sure that your percentages add to 100.
| Source | Percentage |
| Fossil fuel production and use | |
| Food production | |
| Solid and liquid wastes | |
| Halocarbons |
See Appendix 3 for Exercise 19.3 answers.
Media Attributions
- Figure \(\PageIndex{1}\): © Steven Earle. CC BY. Based on World Population Annual 12000 years data .
- Figure \(\PageIndex{2}\): © Steven Earle. CC BY. Based on data from NASA.
- Figure \(\PageIndex{3}\): © Steven Earle. CC BY. Based on data in IPCC AR 5, 2014.


9.21: Reading- Signs of an Eruption
In 2005 USGS geologist Chris Newhall made a list of the six most important signs of an imminent volcanic eruption. They are as follows:
- Gas leaks — the release of gases (mostly H 2 O, CO 2 , and SO 2 ) from the magma into the atmosphere through cracks in the overlying rock
- Bit of a bulge — the deformation of part of the volcano, indicating that a magma chamber at depth is swelling or becoming more pressurized
- Getting shaky — many (hundreds to thousands) of small earthquakes, indicating that magma is on the move. The quakes may be the result of the magma forcing the surrounding rocks to crack, or a harmonic vibration that is evidence of magmatic fluids moving underground.
- Dropping fast — a sudden decrease in the rate of seismicity, which may indicate that magma has stalled, which could mean that something is about to give way
- Big bump — a pronounced bulge on the side of the volcano (like the one at Mt. St. Helens in 1980), which may indicate that magma has moved close to surface
- Blowing off steam — steam eruptions (a.k.a. phreatic eruptions ) that happen when magma near the surface heats groundwater to the boiling point. The water eventually explodes, sending fragments of the overlying rock far into the air.
Contributors and Attributions
CC licensed content, Shared previously
- 4.5 Monitoring Volcanoes and Predicting Eruptions. Authored by : Steven Earle. Provided by : BC Campus. Located at : https://opentextbc.ca/geology/chapter/4-5-monitoring-volcanoes-and-predicting-eruptions/ . Project : Physical Geology. License : CC BY: Attribution


2.10: Microsoft Word
- Use Microsoft Word to create a visually appealing and accessible document
This page provides guidance on how to use key features of Microsoft Word to create your business communication. If you can envision the page layout, there is a way for the software to do it. If the instructions are not found below, it is likely someone has made a YouTube video of just what you want to do.
Getting Started in Word
The following six-minute video demonstrates how to use Microsoft Word to create a letter. Software versions do change over time, so worry less about the details and more about how the information is entered. Each new version retains the core features of all versions before it, so it will still be easy to use with a few adjustments. There are some differences when working on a PC versus an Apple product. These instructions are for the PC, but the basic idea is the same.
Line and Paragraph Spacing in Word
Business communications, unlike writing intended for an academic setting, use single line spacing, as seen in the images below. Both kinds of documents use a left alignment or justification, while business documents might have an unjustified right edge (which leaves line-ends ragged) or fully justified blocks, which are nice and neat on both right and left but might do some funky things with internal spacing to achieve the neatness.
| Avoid | Try | ||
|---|---|---|---|
|
|
To change the line spacing, select the lines you want to change. Then go to the Home tab and find the line spacing button in the Paragraph section. The line spacing button triggers a dropdown menu from which you can select a line spacing.
The numbers in the line spacing dropdown menu reflect how much the line spacing will increase relative to the size of the line. For example, a line spacing of 1 means the line spacing will be as large as one line, a line spacing of 2 means the line spacing will be the size of two lines, and so on.
You may have noticed that by default, Word adds a space between each paragraph. You can change the size of that space or eliminate it completely by changing the paragraph spacing. There is an option to Add Space Before Paragraph or Remove Space After Paragraph in the line spacing menu.
Headers and Footers
Headers and footers are the top-most and bottom-most sections of the document respectively. They are formatted as separate sections from the main document, and are often used to hold footnotes, page numbers, titles, and other information.
Your company likely has conventions and rules in place for the content of headers and footers and also for where page numbers are to be placed. There are, a few standard guidelines to note:
- Page numbers go in the header or footer. Once you decide where to put the page number, Word will keep track of it for you.
- Business letters do not have headers, and they get page numbers only if they run to three or more pages.
- Footers containing page numbers can start on the first page.
- For headers and footers in reports, see Module 6.
The header and footer controls are in the Header & Footer group of the Insert tab. (Insert > Header & Footer)
The Header and Footer options each contain a dropdown menu. Click one of the Built-in options to insert it, or select Edit Header (or Edit Footer) to edit the header (or footer).
The header/footer will be represented by a dotted line. While in header/footer mode (when the dotted lines are visible), you cannot edit the main text. You can go back to editing the main text by selecting the Close Header and Footer option on the right side of the ribbon.
The information in the header or footer is repeated in every header or footer in the document unless you check “Different First Page,” which allows you to format the first page differently from the rest of your document. You can also create a break in your document where the header and/or footer change to accommodate a new section.
Spell Check and Grammar Check
You’ve probably seen the red wavy lines indicating a misspelled word as you type a document. Sometimes, those red lines are helpful, saving you from making an embarrassing typo. Other times, they are annoying and distracting. In this section, we will cover how Spell Check works and how you can best use it to your advantage.
The spell checker tool compares every word you type against Word’s dictionary or database of words. If a word isn’t in this dictionary, the word is marked misspelled with a red underline. This means words that are correctly spelled may be marked as misspelled, especially brand names, specialized industry or scientific terms, and words in another language. With a little tweaking, you can teach Microsoft Word (and other Office programs) how to spell uncommon words and names you use regularly. That way, genuine mistakes are not lost in a sea of red underlines.
Most word processing software has spell and grammar check automatically turned on, so your errors are marked as you work. However, you can also run spell and grammar check manually by pressing F7, or by locating the Spelling and Grammar Check in the Review tab in the ribbon.
Sentence Length and Word Count
Earlier in this module concise writing was emphasized. How do you know your average sentence length? Word has a feature that helps track the lengths of sentences. Here’s how:
- Go to File > Options > Proofing and look for “When correcting spelling and grammar in Word.”
- Now check the box for “Show readability statistics.”
Some online forms also require you to stick to a specific word count or whatever you enter will be cut off by the form’s programming. In the Review tab in the ribbon, there is an option for word count that shows the number of characters (with or without spaces), words, and lines.
Contributors and Attributions
- Microsoft Word. Authored by : Susan Kendall. Provided by : Lumen Learning. License : CC BY: Attribution
- Line and Paragraph Spacing. Provided by : Lumen Learning. Located at : courses.lumenlearning.com/wmopen-compapp/chapter/line-spacing/. License : CC BY: Attribution
- Lists. Provided by : Lumen Learning. Located at : courses.lumenlearning.com/wmopen-compapp/chapter/lists/. License : CC BY: Attribution
- Spell Check and Dictionaries. Provided by : Lumen Learning. Located at : courses.lumenlearning.com/wmopen-compapp/chapter/microsoft-word-spellcheck-and-dictionaries/. License : CC BY: Attribution
- Headers and Footers. Provided by : Lumen Learning. Located at : courses.lumenlearning.com/wmopen-compapp/chapter/headers-and-footers/. License : CC BY: Attribution
- Microsoft Word 2007 Business Letter Tutorial. Authored by : Mr Blais. Located at : https://youtu.be/tD8Gj7vkh-0 . License : All Rights Reserved . License Terms : Standard YouTube License


7.6: Appendix A - Designing a Database
Designing a Database
Suppose a university wants to create an information system to track participation in student clubs. After interviewing several people, the design team learns that the goal of implementing the system is to give better insight into how the university funds clubs. This will be accomplished by tracking how many members each club has and how active the clubs are. From this, the team decides that the system must keep track of the clubs, their members, and their events. Using this information, the design team determines that the following tables need to be created:
- Clubs: this will track the club name, the club president, and a short description of the club.
- Students: student name, e-mail, and year of birth.
- Memberships: this table will correlate students with clubs, allowing us to have any given student join multiple clubs.
- Events: this table will track when the clubs meet and how many students showed up.
Now that the design team has determined which tables to create, they need to define the specific information that each table will hold. This requires identifying the fields that will be in each table. For example, Club Name would be one of the fields in the Clubs table. First Name and Last Name would be fields in the Students table. Finally, since this will be a relational database, every table should have a field in common with at least one other table (in other words: they should have a relationship with each other).
In order to properly create this relationship, a primary key must be selected for each table. This key is a unique identifier for each record in the table. For example, in the Students table, it might be possible to use students’ last name as a way to uniquely identify them. However, it is more than likely that some students will share a last name (like Rodriguez, Smith, or Lee), so a different field should be selected. A student’s e-mail address might be a good choice for a primary key, since e-mail addresses are unique. However, a primary key cannot change, so this would mean that if students changed their e-mail address we would have to remove them from the database and then re-insert them – not an attractive proposition. Our solution is to create a value for each student — a user ID — that will act as a primary key. We will also do this for each of the student clubs. This solution is quite common and is the reason you have so many user IDs!
You can see the final database design in the figure below:
Student Clubs database diagram
With this design, not only do we have a way to organize all of the information we need to meet the requirements, but we have also successfully related all the tables together. Here’s what the database tables might look like with some sample data. Note that the Memberships table has the sole purpose of allowing us to relate multiple students to multiple clubs.


<IP_ADDRESS>: Background Reading
Background Reading
Tip: Keeping Track of Your Information
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It’s wise to do some more reading about that narrower topic once you have it. For one reason, you probably don’t know much about it yet. For another, such reading will help you learn the terms used by professionals and scholars who have studied your narrower topic. Those terms are certain to be helpful when you’re looking for sources later, so jot them down or otherwise remember them.
For instance, if you were going to do research about the treatment for humans with bird flu, this background reading would teach you that professionals and scholars usually use the term avian influenza instead of bird flu when they write about it. (Often, they also use H1N1 or H1N9 to identify the strain.) If you didn’t learn that, you would miss the kinds of sources you’ll eventually need for your assignment.
Most sources other than journal articles are good sources for this initial reading, including the New York Times or other mainline American news outlets, Wikipedia, encyclopedias for the discipline your topic is in (horticulture for the crabapple bud development topic, for instance), dictionaries for the discipline, and manuals, handbooks, blogs, and web pages that could be relevant.
This initial reading could cause you to narrow your topic further, which is fine because narrower topics lead to greater specificity for what you have to find out.After this upfront work, you’re ready to start developing the research question(s) you will try to answer for your assignment.
Tip: Keeping Track of Your Information
While you are in the background reading phase of your research you will come across a lot of sources and won’t know yet if they will prove useful in the long run. A handy type of software to help you keep track of all your findings is called citation management software. It will also be extremely valuable when it comes to using the resources you end up needing. Three of these tools are available for free to OSU students, staff and faculty. Learn more about these tools and how to access them.
It’s worth remembering that reading, scanning, looking at, and listening to information resources is very useful during any step of the process to develop research questions. Doing so can jog our memories, give us details that will help us focus, and help us connect disparate information–all of which will help us come up with research questions that we find interesting.


6.4: Woodwinds
Introduction
Woodwind instruments are a family of musical instruments within the more general category of wind instruments. There are two main types of woodwind instruments: flutes and reed instruments (otherwise called reed pipes). What differentiates these instruments from other wind instruments is the way in which they produce their sound.
Modern Orchestra and Concert Band Woodwinds
The modern orchestra’s woodwind section typically includes: flutes, oboes, clarinets, and bassoons. The piccolo, cor anglais, bass clarinet, and contrabassoon are commonly used supplementary woodwind instruments. The section may also on occasion be expanded by the addition of saxophone(s).
The concert band’s woodwind section is typically much larger and more diverse than the orchestra’s. The concert band’s woodwind section typically includes: piccolo, flutes, oboes, B ♭ clarinets, bass clarinets, bassoons, alto saxophones, tenor saxophone, and baritone saxophone. The cor anglais, E ♭ clarinet, alto clarinet, contra-alto clarinet, contrabass clarinet, contrabassoon, and soprano saxophone are also used, but not as frequently as the other woodwinds.
Flutes produce sound by directing a focused stream of air across the edge of a hole in a cylindrical tube. The flute family can be divided into two sub-families: open flutes, and closed flutes.
To produce a sound with open flutes, the player is required to blow a stream of air across a sharp edge that then splits the airstream . This split airstream then acts upon the air column contained within the flute’s hollow causing it to vibrate and produce sound. Examples of open flutes are the transverse flute, panpipes and ocarinas. Ancient flutes of this variety were often made from tubular sections of plants such as grasses, reeds, and hollowed-out tree branches. Later, flutes were made of metals such as tin, copper, or bronze. Modern concert flutes are usually made of high-grade metal alloys, usually containing nickel, silver, copper, or gold.
To produce a sound with a closed flute, the player is required to blow air into a duct. This duct acts as a channel, bringing the air to a sharp edge. As with the open flutes, the air is then split; this causes the column of air within the closed flute to vibrate and produce sound. Examples of this type of flute include the recorder (instrument), and organ pipes.
Flute
Flutes dating back to 43,000 to 35,000 years ago have been found in Germany, but the contemporary mechanism for contemporary flutes was developed between 1832 and 1847.
Oboe
Listen to an oboe concerto.
Oboes first appeared in the mid-seventeenth century and were developed further in the nineteenth century.
English Horn
Listen to the English horn.
The English horn originated in about 1720.
Clarinet
Listen to the clarinet.
The clarinet became a standard orchestral instrument by the 1800s.
Bassoon
Listen to the bassoon.
The bassoon was developed in the first half of the nineteenth century.
Contrabassoon
Listen to the contrabassoon.
The contrabassoon was developed in the mid-eighteenth century.
Contributors and Attributions
- Woodwind instrument. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Woodwind_instrument . License : CC BY-SA: Attribution-ShareAlike
- Flute. Provided by : Wikipedia. Located at : http://commons.wikimedia.org/wiki/File:Flute.jpg . License : CC0: No Rights Reserved
- English Horn picture. Authored by : Hustvedt. Provided by : Wikipedia. Located at : https://commons.wikimedia.org/wiki/File:English_Horn_picture.jpg . License : CC BY-SA: Attribution-ShareAlike
- Clarinet. Authored by : Iunity. Provided by : Wikipedia. Located at : https://commons.wikimedia.org/wiki/File:Clarinet.png . License : CC0: No Rights Reserved
- FoxBassoon. Authored by : Gregory F. Maxwell. Provided by : Wikipedia. Located at : https://commons.wikimedia.org/wiki/File:FoxBassoon.png . License : Other . License Terms : GNU Free Documentation License, Version 1.2
- Contrabassoon2 flipped. Authored by : Mezzofortist; Smurfett. Provided by : Wikipedia. Located at : https://commons.wikimedia.org/wiki/File:Contrabassoon2_flipped.png . License : CC BY-SA: Attribution-ShareAlike
- Clarinet. Authored by : Peter Miller. Located at : https://www.flickr.com/photos/pmillera4/16251498490/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- Mozart - Concerto for Flute and Orchestra G-dur K 313 (285C) Emmanuel Pahud.. Authored by : Andrey Kostyagin. Located at : https://youtu.be/8zPkWVqdJXI?t=52s . License : All Rights Reserved . License Terms : Standard YouTube License
- Mozart Oboe Concerto (Allegro Aperto) - Nicholas Daniel / Bu011blohlu00e1vek / BBC Symphony. Authored by : Nicholas Daniel. Located at : https://youtu.be/DhygJyCvYeA?t=59s . License : All Rights Reserved . License Terms : Standard YouTube License
- Mozart Adagio K580a - English horn Giuliano Giuliani. Authored by : Guido Gualandi. Located at : https://youtu.be/ePFIlpBJq1o?t=1s . License : All Rights Reserved . License Terms : Standard YouTube License
- Bernstein performs Gershwin Rhapsody in Blue 1/2. Authored by : Felipe Ruiz de Chu00e1vez. Located at : https://youtu.be/xxb7yNG0DGc?t=15s . License : All Rights Reserved . License Terms : Standard YouTube License
- Mozart: Bassoon Concerto in B-flat major KV191, 1st movement (Allegro). Authored by : bububasso. Located at : https://www.youtube.com/watch?v=ARkz1oCtncM&feature=youtu.be&t=57s . License : All Rights Reserved . License Terms : Standard YouTube License
- Variations on a Theme of Robert Schumann for Contrabassoon (The Happy Farmer). Authored by : timbauernfeind. Located at : https://youtu.be/KIJrMb7mOVI?t=1s . License : All Rights Reserved . License Terms : Standard YouTube License